Let $T : \mathbb R^n\to \mathbb R^n$ be a linear map. Is it true that $Trace (T)=\dfrac 1{\mu(B)}\int_{x \in B} \langle Tx, x\rangle d\mu(x)$,
where $B$ is the closed unit ball in $\mathbb R^n$ and $\mu$ is the usual Lebesgue measure and $\langle.,.\rangle$ is the usual inner-product on $\mathbb R^n$?
Best Answer
The normalization of the formula is wrong, but let me propose a fix. Let
$$K_n=\int_{B\subset\Bbb{R^n}}x_1^2\,d\mu$$
be the integral we would get with $T=e_{11}$, the matrix with $1$ at position $(1,1)$ and zeros elsewhere. I think that we then have for all linear transformations $T:\Bbb{R}^n\to\Bbb{R}^n$ that $$ tr(T)=\frac1{K_n}\int_{B}\langle Tx,x\rangle\,d\mu. $$ I will think of $T$ as a matrix (use the natural basis). We can equally well use the symmetrized version of $T$, i.e. $(T+T^t)/2$ because the quadratic form $\langle Tx,x\rangle$ stays the same, and the symmetrized version shares the same trace.
Given that $T$ is symmetric the rest is easy. By linear algebra there exists an orthogonal matrix $P$ such that $P^tTP=D$ is a diagonal matrix. Furthermore, $P^t=P^{-1}$ and $\det P=1$. Also, the linear substitution $x\mapsto Px$ preserves the unit ball, because $P$ is length preserving. What this means is that it suffices to prove the formula for a diagonal matrix $D=diag(d_1,d_2,\ldots,d_n)$.
Obviously $$I_i=\int_{B\subset\Bbb{R^n}}x_i^2\,d\mu=K_n$$ for all $i=1,2,\ldots,n$ by the symmetries of the sphere. By linearity of the integral we then get that $$ \begin{aligned} \int_{B\subset\Bbb{R^n}}\langle Dx,x\rangle\,d\mu &=\int_{B\subset\Bbb{R^n}}(d_1x_1^2+d_2x_2^2+\cdots+d_nx_n^2)\,d\mu\\ &=\sum_{i=1}^nd_i I_i\\ &=tr(D) K_n, \end{aligned} $$ and we are done.
In other words, instead of the measure of the $n$-dimensional ball you should factor out the integral of $x_1^2$ over that ball. I'm sure the exact value of $K_n$ is known. May be a friendly physicist calculated the moments of inertia of the $n$-dimensional homogeneous ball? Or a probability person has calculated the expected value of $x_1^2$ of a random point uniformly distributed over that ball?